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I posted a similar question long time ago, but after working on it I am still unable to arrive at a solution.

Let's have a group of linear transformations (rotations in thexy-plane):

[tex]R_\theta=\{ (\begin{array}{ccc} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}) \\ : \\ \theta \in [0,2\pi] \}[/tex]

The question is: How can I construct an orthogonal curvilinear coordinates system, in which the parameter [itex]\theta[/itex] works as one coordinate?

What I am supposed to get as a result are essentially the equations defining the cartesian-to-polar transformation.

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My attempt:

Observe that given any vectorx, the orbit [tex]R_{\theta}(\mathbf{x})[/tex] is a parametric curve which is obviously a circle.

The (gradient) vectors

[tex]e_\theta=\frac{\partial R_{\theta}(\mathbf{x})}{\partial \theta}[/tex] are tangent to the curve, so if we consider their orthogonal complement [tex]e_\theta^*[/tex] (which is easy to find), we have already found a family oflocalorthogonal bases.

How can I continue from this point???

I am supposed to get: [itex]r = (x^2 + y^2)^{1/2}[/itex] and [itex]\theta = atan2(y/x) [/itex], but I don't know how to arrive at that.

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# Polar coordinates: derivation from rotation group

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